The evolution of electric field energy is shown in Fig. REF , from which it can be seen that the numerical and theoretical solutions [1]} fit well,
and the uncertainties of electric energy propagate in the similar pattern as mean field.
Fig. REF  provides the particle distribution function over the phase space \((x,u)\)  at \(t=70\) .
The swirling pattern is clearly identified in both expectation and variance values.
More fine structures can be seen in the standard deviation, which provides a clearer way to quantify the stochastic evolution of particles.
